An aerospace engineer is designing a high-precision optical filter for a satellite sensor in the shape of a circular sector. The filter requires a special vacuum sealant along its entire perimeter. The cost of this sealant is £4.50 per millimetre. One prototype filter has a radius of 6 mm and a central angle of 1.2 radians.
(i) Calculate the area of this prototype filter.
(ii) Determine the total cost of the sealant required for this prototype.
The engineer designs a new filter with a fixed area of 25 mm2mm^2mm2.
(i) Show that the cost, £CCC, of the sealant required for this new filter is given by
C=9(25r+r) C = 9\left(\frac{25}{r} + r\right) C=9(r25+r)where rrr is the radius measured in millimetres.
(ii) Find the value of rrr for which the cost of the sealant is minimized, and justify that your answer corresponds to a minimum cost.
317 exam-style questions on AQA A Level Maths 1.8 E: Trigonometry, covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context. Each one has a worked solution and a mark scheme showing where the marks go.